OFFSET
1,3
COMMENTS
Adding a loop at the root makes the graph 3-regular: each vertex has degree 3.
The first differences of this sequence give A091090.
LINKS
Wikipedia, Chip-firing game.
FORMULA
a(n) = Sum_{j=1..m-1} (2^j-1)(b(j)+1), where m = floor(log_2(2n+1)) and b(m)b(m-1)...b(1)b(0) is the binary representation of 2*n+1.
EXAMPLE
If there are four chips at the root, then the root fires and the process ends in a stable configuration.
If there are eight chips at the root, the root can fire three times, sending 3 chips to each child. After this, each child can fire once. After that the root has 4 chips and can fire again. The root fires a total of 4 times.
MAPLE
a:= n-> (l-> add((2^(i-1)-1)*(l[i]+1), i=2..nops(l)-1))(Bits[Split](2*n+1)):
seq(a(n), n=1..70); # Alois P. Heinz, Sep 12 2024
PROG
(Python)
def a(n):
if n <= 2:
return 0
else:
return (n+1) // 2 - 1 + a((n+1)//2 - 1)
print([a(2*n) for n in range(1, 51)])
(Python)
def A376116(n): return (n<<1)-n.bit_count()-n.bit_length() # Chai Wah Wu, Sep 18 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved